So you can always find integer solutions for the ellipse using the single parameter t.

But in rationals x2 - Dy2 = 1, where D is a positive integer will give you hyperbolas, but for each solution for x and y, you get an ellipse, so there is a direct connection between hyperbolas as ellipse generators.

So one can directly connect ellipses to hyperbolas using Pell's Equation as a hyperbola itself in rationals and get Diophantine solutions for the ellipses as well, so the rational hyperbola can generate integer solutions to ellipses, including for the special case D-1, Pythagorean Triples.